OK.

All right.

So with a bit of a different flavor,

so I'm coming from the optimal control perspective.

And compared to what Tobias just did,

I am more interested in a Fokker-Blank equation

with very low regularity in the data.

And the problem looks like this.

So we have an optimal more or less standard optimal control

problem.

So we want to minimize a certain objective function

with a standard tracking term here.

This is for simplicity on the whole time and space cylinder.

We have a classical control cost and two kind

of regularization here.

And the governing equation here is a Fokker-Blank equation

in divergence form.

So I wrote both terms in the divergence form

here on the whole space for simplicity.

And also there's an initial value, of course.

So this is a, as I said, it's a classical Fokker-Blank

equation, so to say, which, as you probably know,

has several connections to stochastic PDEs

or differential equations in general,

because you can, in certain cases,

have an equivalence between the temporal evolution

of a probability density function in the stochastic PDEs

with the solutions of this type of Fokker-Blank equation.

And well, since I'm coming from the classical analysis side,

I used to put this in the divergence form

because it is easier for me and actually also

easier to put in the assumptions which we are going to have.

All right, so the control that we have in the system

is here in this first order term, so in the drift.

And we are going to have it in a way

you can also see this here in the function

that the control is just a temporal function only.

So it's just an, let's say, L2 function from 0 to t.

And we also have the stationary drift vector alpha here.

So the drift direction in space is fixed.

And so to say, we control the amplitude in a certain sense.

And there is a lot of recent work, actually,

quite some papers on control of Fokker-Blank equations.

I just mentioned three exemplarily.

So this is Leich and Gullemi, here, the first one.

This is Breiton, Kunisch, and Piper.

Not Piper, and Pfeiffer, sorry.

And this is Arona and Trölsch, which is very recent.

And compared to their works, we assume what we want to do

is we want to go ahead with very low regularity, especially